1

A modular Takagi-Sugeno-Kang (TSK) system based on a

modified hybrid soft clustering for stock selection

Somayeh Mousavi

Department of Industrial Engineering, Meybod University, Meybod, Iran, e-mail address: mousavi@meybod.ac.ir

Yahyazadeh Blvd., Khorramshahr Blvd., Meybod, Yazd, Iran

Tel: +98(35)3321-2412; Mobile: +98-913-152-7828; Fax: +98(35)3235-3004

Akbar Esfahanipour*

* Corresponding Author

Department of Industrial Engineering and Management Systems, Amirkabir University of

Technology, Tehran, Iran, e-mail address: esfahaa@aut.ac.ir

424 Hafez ave. Tehran, Iran 15914 Tel: +98(21)6454-5300; Mobile: +98-912-347-9906; Fax:+98(21)6695-4569

Mohammad Hossein Fazel Zarandi

Department of Industrial Engineering and Management Systems, Amirkabir University of

Technology, Tehran, Iran, e-mail address: zarandi@aut.ac.ir

424 Hafez ave. Tehran, Iran 15914 Tel: +98(21)6454-5300; Fax:+98(21)6695-4569

2

A modular Takagi-Sugeno-Kang (TSK) system based on a modified hybrid

soft clustering for stock selection

This study presents a new hybrid intelligent system with ensemble learning for stock selection

using the fundamental information of companies. The system uses the selected financial ratios

of each company as the input variables and ranks the candidate stocks. Due to the different

characteristics of the companies from different activity sectors, modular system for stock

selection may show a better performance in comparison with an individual system. Here, a

hybrid soft clustering algorithm is proposed to eliminate the noise and partition the input data

set into more homogeneous overlapped subsets. The proposed clustering algorithm benefits

from the strengths of the fuzzy, possibilistic and rough clustering to develop a modular system.

An individual Takagi-Sugeno-Kang (TSK) system is extracted from each subset using an

artificial neural network and genetic algorithm. To integrate the outputs of the individual TSK

systems, a new weighted ensemble strategy is proposed. The performance of the proposed

system is evaluated among 150 companies listed on Tehran Stock Exchange (TSE) regarding

information coefficient, classification accuracy and appreciation in stock price. The

experimental results show that the proposed modular TSK system significantly outperforms the

single TSK system as well as the other ensemble models using different decomposition and

combination strategies.

Keywords: intelligent modular systems; ensemble learning; hybrid rough-fuzzy clustering; TSK

fuzzy rule-based system; stock selection; Tehran Stock Exchange (TSE).

1. Introduction

The fund allocation problem involves two stages, the asset selection, and the asset allocation. In the first stage,

the objective is to select some attractive and valuable assets as the potential candidates for portfolio

composition. In the second stage, the objective is to determine portfolio weights of the selected assets to achieve

a series of risk-return considerations [1]. Similarly, in the stock portfolio management, one should select a

universe of stocks before running a stock portfolio optimization model to determine optimal portfolio weights

[2]. Those selected stocks have the best chances of capital appreciation in a long or intermediate time horizon.

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There are two approaches widely used by academicians and market professionals for decision making in

stock exchange: technical analysis and fundamental analysis. The fundamental analysis involves a detailed study

of a company’s financial status using the financial ratios and other fundamentals of the company to predict the

future stock price movements. In the fundamental analysis, the main concern is the company’s financial health.

Traders often use this approach to predict the stock price over a long-term investment horizon. On the other

hand, the technical analysis uses the data related to the past behavior of a stock price and volume data series to

forecast the future. Traders use this approach for short-term investment horizons. They deal with the stock

timing and not the company’s financial health [3-7]. Since the fundamentals have a stronger relationship to the

price movement in the longer horizons, the stock selection stage should be designed based on the fundamental

analysis.

This study tries to develop a stock selection system based on the fundamental analysis. The system uses the

selected financial ratios and fundamental data of each company as the input variables and ranks the candidate

stocks based on the fundamental data. However, the future performance of the companies may follow different

patterns due to different fundamental characteristics as well as activity sectors of the companies [8, 9]. For

example, the companies with high inventory turnover usually have lower current ratio than the companies in

other activity sectors. A low current ratio is not always an indicator of poor liquidity performance and should be

compared with current ratios of the other companies with the similar inventory turnover. Therefore, it seems a

modular system for stock selection may show a better performance in comparison with an individual system

[10]. Furthermore, based on the principle of divide and conquer, the complexity of the whole data space is

reduced by modularity, which leads to some more homogeneous data spaces [11].

In general, three main steps should be done to develop a modular system with ensembles. In the first step,

the training data set is partitioned into some smaller data regions. In the second step, an individual learner is

developed for each data region, separately. In the third step, the outputs of the individual learners are combined

to determine the final output of the modular system using an ensemble strategy. In this study, a hybrid rough-

fuzzy noise rejection clustering algorithm is proposed to determine the overlapping data regions of the modular

system and remove the noise data, simultaneously. Then, an individual Takagi-Sugeno-Kang (TSK) fuzzy rule-

based system is developed for stock selection in each data region separately. Finally, the outputs of the

individual systems are combined to derive the ultimate result using our proposed ensemble strategy.

The main purpose of this paper is to construct an accurate and interpretable stock selection system for

portfolio managers. The system ranks the universe of stocks and selects a set of stocks that are likely to have the

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best chances of capital appreciation in the subsequent period. For this purpose, we propose an ensemble learning

model to develop a modular stock selection system based on our proposed clustering. Here we describe the

novelties of this study from two perspectives. The first one regards the applied method to modularize the system

and the ensemble strategy. This study proposes a hybrid rough-fuzzy noise rejection clustering algorithm to

partition a noisy data set into some overlapping partitions without noise and outliers. The proposed model also

develops a new weighted ensemble strategy to aggregate the outputs of the modules. The second one relates to

the development of a modular system for stock selection problem through the ranking of the stocks. From the

best of the authors’ knowledge, this is the first study that develops a TSK system for stock selection problem

which applies an ensemble learning model.

The rest of the paper is organized as follows. The next section provides a brief review of the previous works

related to our study. The third section explains the proposed algorithm for hybrid rough-fuzzy noise-rejection

clustering. Section 4 presents a comprehensive description of the proposed ensemble learning model. The fifth

section describes the implementation of the proposed model to develop a modular system for stock selection on

Tehran Stock Exchange as well as the computational results. Finally, section 6 reports some concluding

remarks.

2. Related Works

According to the principle of divide and conquer, a complex computational learning can be simplified by

dividing the learning task among some experts and then combining the solutions of the experts, which is said to

constitute a committee machine [10]. Modular systems are one of the architectural types of committee machines

with local accuracy perspective [12]. In the modular systems, separate learners are developed and applied to

different regions of the problem domain. The regions of interest are determined first, and an individual learner is

then developed for each region. With this architecture, an important issue is the identification of the best regions

to be considered. The regions can be defined based on expert opinion [13] or using purely mechanical means

like clustering [14].

2.1. Related works on clustering algorithms

Many researchers have used hard clustering to determine the data regions of a modular system [15-17].

However, the boundaries between the adjacent data regions may be unclear, and a data instance may not

completely belong to only one cluster. Therefore, hard clustering is too restrictive in partitioning. The soft

clustering algorithms using fuzzy and rough set theories is less restrictive than the hard clustering by permitting

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an instance to belong to multiple clusters. While the fuzzy clustering can efficiently handle the overlapping

clusters [18], it can be too descriptive with potentially a list of possible memberships for an individual object

[19]. In this way, the data set may not be partitioned into some homogenous data spaces with less complexity.

Therefore, most of the previous studies related to the modular systems development with fuzzy clustering

applied the traditional fuzzy clustering in which each sample is assigned to only one cluster based on the

maximum membership [20]. Rough clustering, as another soft clustering approach, allows an object to belong to

multiple clusters. In rough clustering, representation of the clusters is based on the lower and upper bounds

using rough set theoretic properties. Lingras and West [21] believe that the lower and upper bound

representation of a cluster is more concise than the detailed and descriptive list of membership values. They

suggest rough c-means clustering as the first rough clustering algorithm. In this algorithm, the clusters are

represented by the crisp lower and upper approximations. The lower approximation contains objects that are

members of the cluster with certainty (probability = 1), while the upper approximation contains objects that are

members of the cluster with non-zero probability (probability > 0). However, the rough clusters don't determine

the similarity and closeness of the instances to the cluster prototypes [22].

In the last decade, intensive works have been done for the hybridization of rough and fuzzy clustering to

integrate the advantages of both fuzzy sets and rough sets [18, 22-25]. Hybrid rough-fuzzy c-means clustering is

proposed by Mitra et al. in 2006 for the first time [22]. In their proposed clustering, each cluster consists of a

fuzzy lower approximation and a fuzzy boundary. A hybrid clustering with the crisp lower approximation and

the fuzzy boundaries is proposed by Maji and Pal [24]. Furthermore, a rough-fuzzy possibilistic c-means

clustering is extended to make the previous hybrid clustering [24] robust in the presence of noise and outliers

[18]. Many research fields benefit from the application of the hybrid rough-fuzzy clustering such as

bioinformatics and medical imaging [26, 27] and text-graphics segmentation [28]. Recently, the ensemble-based

rough-fuzzy clustering is extended for categorical data with different dissimilarity measures [29, 30]. However,

the mentioned hybrid clustering algorithms suffer from some weak points that are explained in section 3. This

study proposes a new hybrid clustering algorithm which tries to overcome the weakness of these algorithms for

developing a modular system. The proposed algorithm benefits the strengths of the fuzzy, possibilistic and

rough clustering approaches, while lacks their weak points for developing a modular system.

2.2. Related works on ensemble strategies

The literature on ensemble learning has shown that the ensemble can outperform single predictors in many cases

[31-33]. Additionally, ensembles of expert systems have already been successfully applied in financial

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forecasting [34-43]. There are different ensemble strategies in the literature, including simple majority vote,

simple averaging, weighted averaging, reliability based strategies, Bayesian methods and stacking [44, 45].

Among the mentioned ensemble strategies, the simple majority vote and the simple averaging of the baseline

classifiers have shown relatively poor performance in different fields including financial time series prediction

[12, 40]. On the other hand, weighted averaging and stacking strategies have been applied with great success

over the last few years [13, 41, 46-49]. In stacking, a high-level base learner is developed to combine the lower

level base learners, while in the weighted averaging the base learners are combined with different weights. In

these ensemble approaches, the integration of the base learners is done using a weighted least squares algorithm

[46, 48, 49], generalized regression neural networks [47], particle swarm optimization [41] and genetic fuzzy

systems [13]. Lv et al. [48] showed that using the fuzzy memberships to different data partitions improves the

accuracy of the ensemble model. Considering the results of the mentioned studies and our proposed clustering

method, in this work we introduce a new weighted ensemble that uses the rough fuzzy memberships of the data

partitions.

2.3. Stock selection based on the fundamental analysis

Fundamental analysis has been widely used for stock selection in the stock portfolio management. The stock

selection models based on Fundamental analysis include, PROMETHEE decision making model [50],

generalized data envelopment analysis model [1], multiple attributes decision making (MADM) model [51],

Probit and Tobit based models [52] and also a continuous time model for active portfolio management [53].

However, soft computing models seem to be more appropriate for modeling the noisy, nonlinear and complex

behavior of the stock markets [54-58]. In the literature, there are some studies on the soft computing methods

such as artificial neural networks [3, 59], evolutionary algorithms [60-63], support vector machines [64] and

fuzzy logic [65-68] for stock selection problem. This study proposes a hybrid genetic fuzzy system to select the

best stocks for considering in the portfolio composition. The TSK fuzzy rule-based systems have shown good

capability for modeling the nonlinear dynamic systems in many fields including the short-term stock trend

prediction [69-71]. In this paper, we intend to evaluate the performance of TSK systems in the stock ranking and

selection problem over the longer investment horizons. The structure and parameter identification phases of the

TSK systems are done using artificial neural networks (ANN) and genetic algorithms (GA), respectively. Table

1 compares this study with the previous researches for stock selection based on the fundamental analysis using

soft computing methods.

Please insert Table 1 about here.

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3. The Developed Hybrid Rough-Fuzzy Noise-Rejection Clustering Algorithm

The hybrid rough-fuzzy clustering incorporates fuzzy membership value in the rough clustering framework.

Rough-fuzzy c-means (RFCM) algorithm was proposed by Mitra et al. [22] for the first time. This algorithm

partitions a set of N objects 𝑋 = {𝑥1, … , 𝑥𝑗 , … , 𝑥𝑁} into c rough clusters (𝑈𝑖) with a fuzzy lower approximation

and a fuzzy boundary by minimizing the objective function JRFCM as (1) subject to ∑ 𝑢𝑖𝑗𝑐𝑖=1 = 1 for all 𝑗 =

1,2, . . . , 𝑁.

𝐽𝑅𝐹𝐶𝑀 =∑𝐽𝐶𝑖

𝑐

𝑖=1

𝐽𝐶𝑖 = {

𝑤𝑙𝑜𝑤 × 𝐴𝑖 + 𝑤𝑏𝑜𝑢𝑛𝑑 × 𝐵𝑖 , 𝑖𝑓 𝐴𝑈𝑖 ≠ ∅, 𝐵𝑈𝑖 ≠ ∅

𝐴𝑖, 𝑖𝑓 𝐴𝑈𝑖 ≠ ∅, 𝐵𝑈𝑖 = ∅

𝐵𝑖 , 𝑖𝑓 𝐴𝑈𝑖 = ∅, 𝐵𝑈𝑖 ≠ ∅

𝐴𝑖 = ∑ 𝑢𝑖𝑗𝑚‖𝑥𝑗 − 𝑣𝑖‖

2

𝑥𝑗∈𝐴𝑈𝑖

𝐵𝑖 = ∑ 𝑢𝑖𝑗𝑚‖𝑥𝑗 − 𝑣𝑖‖

2𝑥𝑗∈𝐵𝑈𝑖

.

(1)

where 𝑣𝑖 is the center of cluster 𝑈𝑖, ‖. ‖ is the distance norm, 𝑢𝑖𝑗 is the membership of 𝑥𝑗 to cluster 𝑈𝑖 and

1 ≤ 𝑚 < ∞ is the fuzzifier in the fuzzy set theory. 𝐴𝑈𝑖 and 𝐴𝑈𝑖 are the lower and the upper approximations of

𝑈𝑖 and 𝐵𝑈𝑖 = 𝐴𝑈𝑖 − 𝐴𝑈𝑖 denotes the boundary region of the rough cluster 𝑈𝑖. The terms 𝐴𝑖 and 𝐵𝑖 represent

the weighted within-groups sum of squared errors for the lower approximation and boundary of rough clusters,

respectively. The parameters 𝑤𝑙𝑜𝑤 and 𝑤𝑏𝑜𝑢𝑛𝑑 are the relative importance of the lower approximation and the

boundary regions.

According to the definitions of lower approximation and boundary of rough sets by Pawlak, if an object is

the member of a cluster’s lower approximation, it definitely belongs to that cluster and it cannot be a member of

its boundary or the other clusters [72].

𝐼𝐹 𝑥𝑗 ∈ 𝐴𝑈𝑖 𝑇𝐻𝐸𝑁 𝑥𝑗 ∉ 𝐵𝑈𝑘 , ∀𝑘 𝐴𝑁𝐷 𝑥𝑗 ∉ 𝐴𝑈𝑘 , ∀𝑘 ≠ 𝑖

Maji and Pal [24] claimed that based on these definitions, the objects in the lower approximation should

have a similar influence on only their corresponding cluster regardless of their similarity with their

corresponding clusters and the other clusters. They proposed a hybrid rough-fuzzy clustering with the crisp

lower approximation and the fuzzy boundaries. In this case, they reduced 𝐴𝑖 to (2).

𝐴𝑖 = ∑ ‖𝑥𝑗 − 𝑣𝑖‖2

𝑥𝑗∈𝐴𝑈𝑖 . (2)

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They incorporated possibilistic c-means (PCM) into their previous model to develop a more robust

clustering algorithm in the presence of noise and outliers [18]. In their proposed rough-fuzzy possibilistic c-

means algorithm, they calculated 𝐴𝑖 similar to their previous work and changed 𝐵𝑖 as (3).

𝐵𝑖 = ∑ {𝑎(𝜇𝑖𝑗)𝑚1 + 𝑏(𝜈𝑖𝑗)

𝑚2}‖𝑥𝑗 − 𝑣𝑖‖2

𝑥𝑗∈𝐵𝑈𝑖+ 𝜂𝑖 ∑ (1 − 𝜈𝑖𝑗)

𝑚2 𝑥𝑗∈𝐵𝑈𝑖. (3)

where 𝜇𝑖𝑗 is the probabilistic membership of 𝑥𝑗 to 𝑈𝑖 as that in FCM and 𝜈𝑖𝑗 is the possibilistic membership as in

the PCM. The constants a and b determine the relative importance of the probabilistic and possibilistic

memberships, respectively. The objective function of their proposed clustering algorithm is minimized when,

𝜇𝑖𝑗 = (∑ (‖𝑥𝑗−𝑣𝑖‖

2

‖𝑥𝑗−𝑣𝑘‖2)

2

𝑚1−1𝑐𝑘=1 )

−1

.

(4)

𝜈𝑖𝑗 =1

1+(𝑏‖𝑥𝑗−𝑣𝑖‖

𝜂𝑖)

1𝑚2−1

. (5)

Also, a rough possibilistic type 2 fuzzy c-means (RPT2FCM) clustering algorithm is proposed by Sarkar et

al. [73]. The RPT2FCM algorithm is so similar to the rough-fuzzy possibilistic c-means algorithm proposed by

Maji and Pal [18]. The only difference is in the probabilistic memberships. In [73], the probabilistic

memberships of Equation 3 are type 2 fuzzy membership values to handle some other various subtle

uncertainties in the overlapping areas.

Although the lower approximation members of rough clusters definitely belong to their corresponding

clusters, it is unreasonable to impose the same weight for all objects of a lower approximation [74]. We believe

that different objects of a lower approximation should have different weights based on the proximity to their

corresponding cluster prototypes regardless of the other prototypes. This is consistent with the above definition

of lower approximation by Pawlak [72]. In this paper, we propose a hybrid rough-fuzzy noise rejection

clustering (RFNRC) algorithm to resolve some of the drawbacks of the previous hybrid c-means algorithms. It

incorporates the fuzzy noise rejection clustering (FNRC) [75] into the rough c-means (RCM) framework. The

FNRC utilizes FCM and PCM to introduce a more robust clustering algorithm in the presence of noise and

outliers.

In our proposed algorithm, different objects have different weights in determining their prototypes similar to

[22]. However, unlike their work, the weights only depend on the distance of the objects from their

corresponding prototypes which is more consistent with the rough set theory. The main steps of the proposed

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RFNRC algorithm are as follows. The steps 1-3 are designed to define the suitable weighting exponent, the

number of clusters and the initial cluster centers as the preprocessing steps of RFNRC. The fourth step

determines the fuzzy clusters. The steps 5-7 are related to noise rejection from data set. In the 8th step, the PCM

membership values are calculated, and the steps 9-10 determine the rough clusters.

(1) Define the suitable weighting exponent (m). The weighting exponent should be selected far from its both

extremes to guarantee that the cluster validity index in the next step indicates the optimum number of

clusters. According to [76], the suitable weight exponent is a value that makes the trace of the fuzzy total

scatter matrix (ST) equal to z/2.

𝑆𝑇 = ∑ (∑ (𝑢𝑖𝑗)𝑚𝑐

𝑖=1 )𝑁𝑗=1 (𝑥𝑗 − �̅�)(𝑥𝑗 − �̅�)

𝑇 . (6)

𝑧 = trace (∑ [(𝑥𝑗 −1

𝑁∑ 𝑥𝑗𝑁𝑗=1 ) (𝑥𝑗 −

1

𝑁∑ 𝑥𝑗𝑁𝑗=1 )

𝑇

]𝑁𝑗=1 ) .

(7)

where �̅� is the fuzzy total mean vector of the dataset considering the FCM based membership values.

(2) Determine the optimum number of clusters (C) through the original FCM so that the cluster validity index

(8) is minimized. This index determines the optimum number of clusters that maximizes within clusters

compactness and between clusters separation.

𝑆𝑐𝑠 = ∑ ∑ (𝑢𝑖𝑗)𝑚(𝑐

𝑖=1𝑁𝑗=1 ‖𝑥𝑗 − 𝑣𝑖‖

2− ‖𝑣𝑖 − �̅�‖

2) . (8)

(3) Assign the initial cluster centers by the agglomerative hierarchical clustering algorithm. This clustering

algorithm puts each of the n data instances in an individual cluster. Then, two or more clusters are merged

using a matrix of dissimilarities, until the required number of clusters (C) are reached. This step prevents

our proposed algorithm to converge to a local extreme.

(4) Identify the initial fuzzy cluster prototypes using the original FCM.

(a) Compute the matrix of membership degrees:

𝑢𝑖𝑗 = (∑ (‖𝑥𝑗−𝑣𝑖‖

2

‖𝑥𝑗−𝑣𝑘‖2)

2

𝑚−1𝐶𝑘=1 )

−1

. (9)

(b) Update the cluster centers:

𝑣𝑖 =∑ (𝑢𝑖𝑗)

𝑚𝑥𝑗

𝑁𝑗=1

∑ (𝑢𝑖𝑗)𝑚𝑁

𝑗=1

. (10)

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(c) Repeat (a) and (b) until reaching convergence in cluster centers, i.e., 𝑣𝑖(𝑡)− 𝑣𝑖

(𝑡−1) < 휀, where t is the

iteration number of the FCM algorithm.

(5) Calculate the resolution parameter of PCM (𝜂𝑖). The value of 𝜂𝑖 determines the distance that the

membership value of a point in the cluster i becomes 0.5 and is chosen based on the desired “bandwidth”

of the possibilistic membership distribution for each cluster [77]. It is assumed that the data in each cluster

follows a Gaussian distribution and ‖𝑥𝑗 − 𝑣𝑖‖ 𝜎𝑖⁄ has a chi-square distribution, with degrees of freedom

equivalent to the number of features in each data instance. Therefore, the resolution parameter can be

calculated as [75]:

𝜂𝑖 =

𝑚𝑒𝑑𝑖𝑎𝑛 (‖𝑥𝑗−𝑣𝑖‖)

𝑥𝑗∈𝑈𝑖

𝜒0.52 . (11)

where χ2 is the chi-square value.

(6) Calculate the cutoff distance (𝑢𝐹𝐶𝑐𝑢𝑡2) to detect the noise and outliers [75].

𝑢𝐹𝐶𝑐𝑢𝑡2 = 𝜂𝑖𝜒�̂�

2 . (12)

where �̂� is the percentage of inliers in the data. The number of outliers is estimated based on W index.

𝑊𝑗 = ∑ ‖𝑥𝑗 − 𝑣𝑖‖𝑪𝒊=𝟏 . (13)

This index sums the distance of the data instance to all cluster centers. The data instances with large values of

Wj are considered as outliers. The threshold for outliers depends on the upper and lower bounds of the data and

is selected according to the trace of 𝑊 index. This step is designed to find the outliers, i.e., the data instances

which are too far from all cluster centers.

(7) Remove the noise data and outliers. If ‖𝑥𝑗 − 𝑣𝑖‖ > 𝑢𝐹𝐶𝑐𝑢𝑡2

, then the data instance is recognized as noise,

and it takes a zero membership to the cluster.

(8) Compute the membership matrix of the remaining data using PCM membership value [77]:

𝑢𝑖𝑗 =1

1 + (‖𝑥𝑗 − 𝑣𝑖‖

𝜂𝑖)

1𝑚−1

. (14)

(9) Assign each data instance to the lower approximation of a cluster or the boundaries of multiple clusters, by

the following procedure:

(a) Assign the instance (xj) to the upper bound of the cluster k (𝐴𝑈𝑘), based on the maximum membership.

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𝑥𝑗 ∈ 𝐴𝑈𝑘 , 𝑘 = arg 𝑖max𝑖=1,2,…,𝐶(𝑢𝑖𝑗) . (15)

The instance should be assigned to the upper bound and boundary of two or more clusters in the case of ties

in the maximum membership.

(b) For each cluster i, i=1, 2, ...,C, i≠k, IF 𝑢𝑘𝑗 − 𝑢𝑖𝑗 < 𝛿 Then 𝑥𝑗 ∈ 𝐴𝑈𝑖 and 𝑥𝑗 ∈ 𝐵𝑈𝑖 ,

where 𝛿 is a small threshold value that determines the overlapping degree of the adjacent clusters.

(c) IF 𝑥𝑗 ∉ 𝐴𝑈𝑖 , 𝑖 = 1,2, . . . , 𝐶, 𝑖 ≠ 𝑘 Then 𝑥𝑗 ∈ 𝐴𝑈𝑘 Else 𝑥𝑗 ∈ 𝐵𝑈𝑘.

(10) Compute the new cluster centers as Equation 16.

𝑣𝑖 = {

𝑤𝑙𝑜𝑤 × 𝐶1 + 𝑤𝑏𝑜𝑢𝑛𝑑 × 𝐷1, 𝑖𝑓 𝐴𝑈𝑖 ≠ ∅, 𝐵𝑈𝑖 ≠ ∅

𝐶1, 𝑖𝑓 𝐴𝑈𝑖 ≠ ∅, 𝐵𝑈𝑖 = ∅

𝐷1, 𝑖𝑓 𝐴𝑈𝑖 = ∅, 𝐵𝑈𝑖 ≠ ∅

𝐶1 =∑ 𝑢𝑖𝑗

𝑚𝑥𝑗𝑥𝑗∈(𝐴𝑈𝑖)

∑ 𝑢𝑖𝑗𝑚

𝑥𝑗∈(𝐴𝑈𝑖)

𝐷1 =∑ 𝑢𝑖𝑗

𝑚𝑥𝑗𝑥𝑗∈(𝐵𝑈𝑖)

∑ 𝑢𝑖𝑗𝑚

𝑥𝑗∈(𝐵𝑈𝑖)

.

(16)

According to the rough set theory, the objects of the lower approximation should have much more influence to

their cluster prototypes. Therefore, 𝑤𝑙𝑜𝑤 should be much more than 𝑤𝑏𝑜𝑢𝑛𝑑, i.e., 0 < 𝑤𝑏𝑜𝑢𝑛𝑑 < 𝑤𝑙𝑜𝑤 <

1,𝑤𝑙𝑜𝑤 + 𝑤𝑏𝑜𝑢𝑛𝑑 = 1.

(11) Repeat steps 8-10 until convergence, i.e., there are no more new assignments.

The strengths of our proposed RFNRC algorithm are in six aspects. First, from the compatibility with the

centroid point of view, possibilistic memberships of PCM correspond more closely to the notion of typicality

[77]. Unlike FCM, there is no constraint on the memberships of PCM (i.e., ∑ 𝜇𝑖𝑗 = 1, ∀𝑗𝐶𝑖=1 ). Therefore, the

prototypes of PCM are attracted toward dense regions in the feature space, regardless of the locations of the

other prototypes. Second, using FCM and agglomerative hierarchical clustering at the first steps avoids the

problem of the coincident clusters of PCM. The FCM and PCM algorithms have been previously integrated to

avoid the problems of noise sensitivity of the FCM and the coincident clusters of PCM [18]. Third, the steps 5-7

of the proposed algorithm makes the algorithm more robust in the presence of noise and outliers. The powerful

ability of these steps in noise rejection has been confirmed by Melek et al. [75]. Our proposed algorithm

removes the noise and outliers from the cluster members and, therefore the outliers don’t affect the learning

process of the modules in the modular system. Fourth, the c-means algorithms with random initial centers

12

always converge to a local extreme [78]. Our proposed RFNRC algorithm has overcome this problem using

agglomerative hierarchical clustering algorithm. Fifth, it uses two preliminary steps to identify the suitable

weight exponent and the optimum number of clusters which makes our algorithm more efficient. Above all, the

previous hybrid clustering algorithms allow only two overlapping clusters [18, 22, 24]. However, three or more

overlapping clusters are possible in clustering of real data sets. Step 9 (parts a &b) of our proposed RFNRC

handles multiple overlapping clusters as well as two overlaps.

4. The Proposed Ensemble Learning Model for Stock Selection

This section describes the general architecture of our proposed model to develop a modular TSK system with

weighted ensemble strategy for stock selection based on fundamental analysis. In this approach, the system uses

fundamental data of companies to predict the future behavior of its stock in the following year and assigns a

score to the stocks. Then the system ranks the universe of stocks based on the assigned scores. Figure 1 shows

the overall framework of the proposed ensemble learning model.

Please insert Fig. 1 about here.

This framework starts with collecting fundamental information of companies along with the data

preprocessing stage to treat the missing variables. We also consider data normalization as one of the necessary

data transformations in the forecasting problems [16]. This study applies the min-max normalization method to

obtain a database with all feature’s values falling in range of 0 and 1. Due to a large number of fundamental

variables as the input variables, the most influential subset of variables are selected by stepwise regression

analysis. Stepwise regression analysis has been used successfully for variable selection in the stock market

forecasting [69-71, 79]. This technique either adds the variables onward or removes the variables backward to

find the best combination of independent variables for forecasting the dependent variable. The following

sections describe the other steps of the proposed model in more detail.

4.1. Data partitioning using the developed hybrid rough-fuzzy noise-rejection clustering

Due to the variability of fundamental properties of the companies within different activity sectors, we believe

that the data partitioning is a necessary task to have the more homogenous subsets to develop a modular system

for stock selection. On the other hand, the fundamental information of the companies generally has some

outliers. Therefore, the training data set is partitioned into multiple overlapping clusters using the proposed

rough-fuzzy noise-rejection clustering algorithm as described in section 3. The proposed algorithm has some

13

advantages for our application. First, because of its noise-rejection property, it can handle the outliers within the

fundamental data set. Second, it benefits from the fuzzy, possibilistic and rough clustering to resolve the

ambiguity in assigning the objects to the modules. The training patterns of the modules are unique in their lower

approximation, and they are similar to patterns of one or more modules in their overlapping or boundary

regions. Third, this algorithm represents the cluster members using their rough-fuzzy memberships. The third

property is useful for designing of an efficient ensemble strategy as described in section 4.3.

4.2. Generating TSK fuzzy rule-based systems for stock selection

In TSK fuzzy rule-based systems, the knowledge base includes multiple fuzzy rules with crisp functions as the

consequent. For the first order TSK system with two input variables, the rules are in the form of:

𝐼𝐹 𝐿1 𝑖𝑠 𝐹𝑆𝑖1 𝐴𝑁𝐷 𝐿2 𝑖𝑠 𝐹𝑆𝑖2 𝑇𝐻𝐸𝑁 𝑦𝑖 = 𝑎𝑖0 + 𝑎𝑖1𝐿1 + 𝑎𝑖2𝐿2

where 𝐿1 and 𝐿2 are the linguistic variables, 𝐹𝑆𝑖1 and 𝐹𝑆𝑖2 are their corresponding fuzzy sets and 𝑎𝑖0 , 𝑎𝑖1, 𝑎𝑖2 are

the parameters of the system. The system inferences using crisp reasoning. The crisp inference of the system is

determined as the weighted average of the individual rule inferences using (17).

�̂� =∑ 𝐷𝑂𝐹𝑖𝑦𝑖𝑟𝑖=1

∑ 𝐷𝑂𝐹𝑖𝑟𝑖=1

⁄ . (17)

where, r is the number of the rules and 𝐷𝑂𝐹𝑖 is the degree of firing of the ith rule, i.e., the rule’s condition

memberships aggregated by a t-norm operator as (18).

𝐷𝑂𝐹𝑖 = 𝜇𝐹𝑆𝑖1(𝐿1) ∧ 𝜇𝐹𝑆𝑖2(𝐿2) . (18)

where ∧ is a t-norm operator and 𝜇𝐴(𝑥) is the membership degree of x in the fuzzy set A.

In our proposed framework, a unique TSK system is independently developed for each data partition as an

individual learner. Each data partition includes the lower approximation and the boundary of the rough cluster

determined by our proposed RFNRC algorithm. The structure and parameter identification of the TSK systems

are described in the following.

We design the TSK rules in canonical form, where the combination of all the input variables takes place

using the conjunction operator. The structure of the TSK rules in the premises is determined by the grid

partitioning. The quality of the TSK system heavily depends on the partitioning of the input space [80]. We

determine the fuzzy sets for the input variables using the modified Adeli-Hung algorithm (AHA) [71]. This

algorithm includes two stages. The first stage involves clustering of the data instances with a topology and

14

weight change neural network, known as Adeli-Hung clustering. The second stage comprises assigning the

membership functions to the input space [81]. These stages are clearly explained in [82]. In the modified version

of AHA, the input variables are partitioned individually, and the membership functions are determined for

individual variables [71]. The modified AHA assigns symmetric triangular fuzzy sets to the input variables,

where the fuzzy sets of the adjacent labels overlap to some extent, and their vertex points do not cross. These

properties make the system more transparent [83].

The parameters of the linear conclusion of the TSK rules are determined using the genetic algorithm. Here,

GA is used to learn the TSK system because of its flexible and powerful search capability [84], especially in the

case of fuzzy systems learning [85, 86]. This evolutionary algorithm has been successfully applied for both

phases of fuzzy modeling, namely structure and parameter identification [80, 87]. Among the two common

approaches for genetic learning of the rule-based systems, i.e., Pittsburgh and Michigan, we apply the Pittsburgh

approach to learn the TSK fuzzy systems. In the Pittsburgh approach, one individual encodes the whole rule

base of the system. Figure 2 shows the encoding scheme of the TSK consequents parameters as a GA

chromosome.

Please insert Fig. 2 about here.

The GA chromosomes are evaluated by information coefficient (IC) as the fitness function. IC is a

performance measure used for evaluating the forecasting skill of financial analysts. It is an appropriate fitness

measure for ranking and classifying the stocks in the investment universe [60, 88]. Information coefficient

measures the Spearman correlation of the ranking that the model assigns to the stocks and their actual rankings

in the following period.

𝐼𝐶𝑡 =∑ 𝑀𝑅𝑎𝑛𝑘𝑗,𝑡∗𝑅𝑅𝑎𝑛𝑘𝑗,𝑡−(∑ 𝑀𝑅𝑎𝑛𝑘𝑗,𝑡∗∑ 𝑅𝑅𝑎𝑛𝑘𝑗,𝑡)

𝑠𝑗=1

𝑠𝑗=1 𝑛𝑡⁄𝑠

𝑗=1

√(∑ 𝑀𝑅𝑎𝑛𝑘𝑗,𝑡2𝑠

𝑗=1 −(∑ 𝑀𝑅𝑎𝑛𝑘𝑗,𝑡)2𝑠

𝑗=1 𝑛𝑡⁄ )∗(∑ 𝑅𝑅𝑎𝑛𝑘𝑗,𝑡2𝑠

𝑗=1 −(∑ 𝑅𝑅𝑎𝑛𝑘𝑗,𝑡)2𝑠

𝑗=1 𝑛𝑡⁄ )

, 𝑡 = 1, 2,… , 𝑇.

(19)

where 𝑀𝑅𝑎𝑛𝑘𝑗,𝑡 is the rank of stock j which is predicted by the model at time t, 𝑅𝑅𝑎𝑛𝑘𝑗,𝑡 is the ranking of the

realized return of stock j at time t, and T is the number of periods in the training set. Our model evaluates the

GA chromosomes by the average information coefficient of the proposed system as (20).

𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛: max{∑ 𝐼𝐶𝑡𝑇𝑡=1 𝑇⁄ } . (20)

4.3. Aggregating the TSK systems based on the proposed weighted ensemble strategy

Our proposed ensemble learning model develops an individual TSK system for each partition of the training

instances. Similar to the training instances, a new instance may belong to multiple partitions with different

15

membership values. Therefore, our model combines the outputs of the corresponding TSK systems using an

ensemble strategy to reach the final score of the new instance. In our proposed modular system, we design a new

weighted ensemble strategy to combine the outputs of the TSK systems. In this design, the weight of each

module is different for each instance and depends on the proximity of the instance concerning the prototype of

module. It uses both rough and possibilistic-fuzzy memberships to calculate the relative importance of the

modules as their weights. The ensemble weight of the module i for the instance 𝑥𝑗(𝐸𝑊𝑖𝑗) is determined by (21).

𝐸𝑊𝑖𝑗 =

{

𝑤(𝑢𝑖𝑗)

𝑤(𝑢𝑖𝑗) + ∑ 𝑤(𝑢𝑘𝑗)𝑘|𝑥𝑗∉𝐴𝑈𝑘

𝑖𝑓 𝑥𝑗 ∈ 𝐴𝑈𝑖

𝑤(𝑢𝑖𝑗)

∑ 𝑤(𝑢𝑘𝑗) + ∑ 𝑤(𝑢𝑘𝑗)𝑘|𝑥𝑗∉𝐴𝑈𝑘𝑘|𝑥𝑗∈𝐵𝑈𝑘

𝑖𝑓 𝑥𝑗 ∈ 𝐵𝑈𝑖

𝑤(𝑢𝑖𝑗)

∑ 𝑤(𝑢𝑘𝑗)𝑘|𝑥𝑗∈𝐴𝑈𝑘+ ∑ 𝑤(𝑢𝑘𝑗) + ∑ 𝑤(𝑢𝑘𝑗)𝑘|𝑥𝑗∉𝐴𝑈𝑘𝑘|𝑥𝑗∈𝐵𝑈𝑘

𝑖𝑓 𝑥𝑗 ∉ 𝐴𝑈𝑖

(21)

where, 𝑤,𝑤 and 𝑤 correspond to the relative importance of the modules regarding the rough partitions. 𝑤 is the

weight of the module if 𝑥𝑗 belongs to its lower approximation. 𝑤 is the weight of the module if 𝑥𝑗 belongs to its

boundary. And 𝑤 is the weight of the module if 𝑥𝑗 does not belong to its partition. Since the instances out of the

upper approximation don’t contribute in the module training, 𝑤 is too small relative to 𝑤 and 𝑤 (0 < 𝑤 ≪ 𝑤 <

𝑤 < 1). The relative importance of the modules depends on the possibilistic-fuzzy membership of the instance

to its partition (𝑢𝑖𝑗), as well.

Figure 3 shows a typical partitioning of the data set in a two-dimensional space. The thickness of the arrows

corresponds to the ensemble weight of the arrowhead’s module. In this figure, 𝑥1 ∈ 𝐴𝑈3 and the ensemble

weight of the third module is very high (𝐸𝑊31 = 𝑤(𝑢31) (𝑤(𝑢31) + 𝑤(𝑢11 + 𝑢21)⁄ )). This instance does not

belong to the other partitions (𝑥1 ∉ 𝐴𝑈1, 𝑥1 ∉ 𝐴𝑈2), therefore the weights which are assigned to their

corresponding modules are very low

(𝐸𝑊11 = 𝑤(𝑢11) (𝑤(𝑢31) + 𝑤(𝑢11 + 𝑢21))⁄ , 𝐸𝑊21 = 𝑤(𝑢21) (𝑤(𝑢31) + 𝑤(𝑢11 + 𝑢21))⁄ ). Another instance,

𝑥2, belongs to the boundaries of 𝑈1 and 𝑈2. Therefore, the ensemble weights of the first and the second modules

are relatively high based on their possibilistic-fuzzy membership values

(𝐸𝑊11 = �̅�(𝑢11) (�̅�(𝑢11 + 𝑢21) + 𝑤(𝑢31))⁄ , 𝐸𝑊21 = (�̅�(𝑢21) �̅�(𝑢11 + 𝑢21) + 𝑤(𝑢31)⁄ )). However, the

weight of the third module is too low (𝐸𝑊31 = (𝑤(𝑢31) �̅�(𝑢11 + 𝑢21) + 𝑤(𝑢31)⁄ )).

Please insert Fig. 3 about here.

16

Consequently, the final score of the stock with fundamental properties represented by 𝑥𝑗 is determined by

(22).

𝑆𝑐𝑜𝑟𝑒𝑥𝑗 = ∑ (𝐸𝑊𝑖𝑗 ∗ �̂�𝑖𝑗)𝐶𝑖=1 . (22)

where �̂�𝑖𝑗 is the output of module i for the instance 𝑥𝑗. Finally, the system selects the stocks according to their

ranked scores at time t.

5. Experimental Results

We implemented our proposed ensemble learning model to develop a modular TSK system for stock selection

among 150 companies with different activity sectors listed on Tehran Stock Exchange (TSE). This section

describes the collected data at first. Then, it reports the implementation results of our proposed model step by

step. Finally, it analyses the performance of the developed modular system and its comparison results.

5.1. Data

In this work, our data comprise of the fundamental data of 150 Iranian companies listed on TSE during 24 fiscal

years to develop a stock selection system for investing in this market. These companies are the most liquid

companies according to six liquidity measures including the number of traded shares, the value of traded shares,

the number of trading days, the number of trades, the average number of shares issued and the company’s value

in a fiscal year. The liquidity measures are aggregated by harmonic mean (Equation 23) to find the most liquid

companies on TSE.

𝑀𝑗 = 𝑁/∑ 𝐼𝑖𝑗𝑁𝑖=1 . (23)

where, 𝑀𝑗 is the jth company’s score, N is the number of indices and 𝐼𝑖𝑗 is the value of the ith index for the jth

company. The companies are selected from the most liquid companies that were active before 2008 from

different activity sectors [89]. We selected thirty-six financial ratios as the potential fundamental variables in

five categories of the profitability, activity, liquidity, leverage and valuation ratios [1, 3, 59, 62, 65, 90]. Table 2

reports a list of the selected financial ratios. We calculated the financial ratios of the Iranian companies listed on

TSE using the financial statements information of the companies.

Please insert Table 2 about here.

The historical data includes the above mentioned financial ratios and the dividend and split-adjusted close

price for the start and end of the companies’ fiscal year from March 20, 1991 to March 19, 2014. We used the

17

extending window approach to define the training and testing periods. This approach uses all the available

historical data to train the system and then applies the trained system on the next period immediately after the

last training period [91]. In this work, we designed five series of experiments to evaluate the evolved system for

stock selection on five fiscal years, March 20, 2009, to March 19, 2014. Table 3 shows the training and the

testing periods of the five series of experiments.

Please insert Table 3 about here.

5.2. Implementation of our proposed ensemble learning model for stock selection

This section reports the implementation results of our proposed ensemble learning model step by step.

Step 1. This step involves the data preprocessing and the variable selection. First, the fundamental data were

studied to handle the missing data and some outliers. Then, the data were normalized. Finally, the most effective

financial ratios were selected using the stepwise regression analysis. We used the statistical software, IBM SPSS

statistics 20 for setting up the regression forecasting model. We considered the thirty-six mentioned financial

ratios as the independent variables and the rate of return as the dependent variable. We set the probabilities of F

statistics to enter and remove a variable at 0.05 and 0.1, respectively. Due to our experiments, five financial

ratios were chosen as the input variables of the TSK systems. The selected ratios are the return on asset (ROA),

total asset turnover, equity ratio, dividend yield and book value per share.

Step 2. We implemented the proposed hybrid rough-fuzzy noise-rejection clustering algorithm to decompose

the training data set into several overlapping subsets (i.e., clusters). At first, the suitable weighting exponent was

selected as m=2.5, which gives a value for the trace of the total scatter matrix equal to 𝑧 2⁄ (Figure 4). Then, the

optimum number of clusters was identified using the cluster validity index. According to Figure 5, the rate of

reduction in the cluster validity index is very high till 4 clusters, and the index gradually decreases till 7 clusters.

We set the number of clusters at C=4 that ensures almost minimum cluster validity index as well as sufficient

training instances to learn the individual TSK systems. The noise data are those with large values of the noise-

rejection index (W) as shown in Figure 6. In our experiments, the threshold was selected as 2 for calculating the

cutoff distance and removing the noise. Then, the FNRC membership degrees were calculated, and the rough-

fuzzy data partitions were determined through the iterative procedure of the proposed algorithm. In this

algorithm, the threshold 𝛿 represents the size of granules of rough-fuzzy clusters. The threshold could be

determined as the median or the mean of the difference of the highest and the second highest memberships of all

the instances to the specified clusters [19, 73]. However, when the distribution of the membership differences is

18

skewed, the median would work better than the mean. Therefore, we have used the following definition for 𝛿

assignment:

𝛿 = 𝑀𝑒𝑑𝑖𝑎𝑛𝑗=1,2,…,𝑁(𝑢𝑖𝑗 − 𝑢𝑘𝑗) . (24)

where 𝑢𝑖𝑗 and 𝑢𝑘𝑗 are the two highest memberships of instance 𝑥𝑗. Based on this definition, the threshold was

set to 𝛿 = 0.22. Consequently, the modular system included four modules with about 600 samples to train the

individual TSK systems.

Please insert Fig. 4 about here.

Please insert Fig. 5 about here.

Please insert Fig. 6 about here.

Step 3. This step resulted in generating the individual TSK systems for each data region, separately. The

structure of the TSK rules' antecedents was specified through AHA. In our design, three fuzzy sets were

allocated to each input variable, interpreted by linguistic labels of low, medium and high. The parameters of the

TSK rules' consequents were learned using GA with Pittsburgh approach. We set the parameters of GA as

shown in Table 4. Crossover and mutation rates and reproduction size are selected based on some primary

experiments. The population size is selected as 100 which showed a better performance in multiple runs. Since

no significant improvements were observed after generation 150, the number of generations is set 200. Four

TSK systems were learned for the four data partitions on the five time frames. Table 5 reports the performance

of the TSK systems in terms of IC on training data subsets. According to this table, the individual TSK systems

could learn a stock selection system with the average IC value of 33.03% in the case of their corresponding

training subsets.

Please insert Table 4 about here.

Please insert Table 5 about here.

Step 4. This step involved aggregation of the outputs of all of the evolved individual TSK systems to reach an

overall stock ranking. The modules' outputs were aggregated using the proposed weighted ensemble strategy,

and then the stocks were ranked according to their scores. According to the preliminary experiments, we set the

𝑤,𝑤 and 𝑤 to 0.8, 0.6 and 0.2, respectively. The next section reports the performance of our proposed ensemble

learning model on the five testing periods and the comparison results.

19

5.3. Performance evaluation of our proposed ensemble learning model

The performance evaluation of our proposed RFNRC-TSK-MBE (rough-fuzzy noise rejection clustering based

modular TSK system with membership based ensemble) is based on information coefficient for stock selection

on TSE. For comparison purposes, we established three alternative models: single TSK, FNRC-TSK-AveE

(FNRC based TSK system with averaging ensemble) and RFNRC-TSK-AveE (RFNRC based TSK system with

averaging ensemble). We designed the single TSK model based on the first and the third steps of our proposed

RFNRC-TSK-MBE. However, the third step in the single TSK system uses the whole training dataset instead of

the training subsets. FNRC-TSK-AveE partitions the dataset using FNRC and then assigns the instances to

clusters based on the maximum membership. This model develops an individual TSK for each disjoint subset

and applies simple averaging as the ensemble strategy. The RFNRC-TSK-AveE uses the same decomposition

and sub-modeling method with RFNRC-TSK-MBE but aggregates the modules by simple averaging ensemble

strategy.

Table 6 reports the experimental results of the four models for stock selection along the five testing periods.

The reported numbers are the average results of 30 independent runs. According to this table, the RFNRC-TSK-

MBE shows a remarkable performance for stock selection. The single TSK system is able to rank the TSE

stocks with IC of 8.5% on average, which is a good performance. However, our proposed modular systems

could outperform the single system. The first modular system extended by FNRC-TSK-AveE model could

improve the ranking ability of the TSK system to IC of 12.74% on average. The ability of the modular system is

further improved by our proposed clustering method. It reaches to 15.24% information coefficient. Above all,

our proposed ensemble method could boost the predictability of the modular system from IC of 15.24% to

18.15%, on average. Similar results are found in all experiments along the five periods. The correlation between

the RFNRC-TSK-MBE model's output and the one year forward return is substantially high over each test

period.

Please insert Table 6 about here.

According to the comparison results, the single TSK model showed the weakest performance for stock

selection. The reason is different fundamental characteristics of the companies in different activity sectors.

Therefore, a modular system works better for such a problem. Furthermore, the FNRC-TSK-AveE got a less IC

in comparison with RFNRC-TSK-AveE. That is because it is difficult to distinguish a certain boundary between

different clusters of data set. Moreover, RFNRC-TSK-MBE arrived at the highest IC value among all the

models. The superiority of the RFNRC-TSK-MBE over the other ensemble models is because of its

20

decomposition and combination methods. However, it is notable that all the developed TSK systems have

shown a good performance in stock selection. As a general guideline, 5% is an acceptable IC value in

investment management [88]. Also, according to a research published by JP Morgan [92], managers with ICs

between 0.05 and 0.15 can achieve significant risk-adjusted excess returns. Therefore, TSK system can model

the stock selection problem properly. The proposed RFNRC-TSK-MBE model could reach IC value of 18.15%

on average which is much more than the other stock selection models in the literature [60, 88, 93, 94]. The

genetic programming model provided by Becker et al. [60] could reach to a maximum IC of 8%. Additionally,

the average IC of 9% was obtained in [88] using grammatical evolution. The authors claimed that 9% is a high

IC value and their models were successful at the stocks' ranking. Also, Gillam et al. [93] studied on the earnings

prediction in a global stock selection model. They could improve the predictability of the model to the IC of 6%.

Finally, we carried out the statistical tests to examine whether the proposed model significantly outperforms

the other three models or not. The results of student t-test are reported in Table 7. According to this table, the

RFNRC-TSK-MBE significantly outperforms the other three models at 99% statistical significance level. This

table shows the impact of modularization (FNRC-TSK-AveE over single TSK) is more than other factors, i.e.,

the clustering and ensemble methods.

Please insert Table 7 about here.

In another experiment, the profitability of our proposed model has been investigated for stock classification.

In our experiment, all the stocks (described in section 5.1) have been classified into two classes. Similar to [3],

we have defined class 1 as the stocks which appreciate in share price to or more than 80% within one year. The

other stocks have been classified as class 2. In this design, the first class constitutes the minority of the data,

while it is our interested class. We have applied the over-sampling technique to deal with the imbalanced data in

training set. In over-sampling, the samples of the rare class are increased by data replication.

The classification performance of our proposed model has been examined in terms of classification

accuracy. For comparison purposes, ANFIS (adaptive neuro-fuzzy inference system) has been implemented on

the same data base to develop a TSK fuzzy rule-based system for stock classification. The performance of

ANFIS for stock classification was previously investigated on Dow Jones Industrial Average (DJIA) market [3].

According to that research, ANFIS outperforms other neural network models, multi-layer perceptron and radial

basis function, in terms of classification accuracy and time complexity. The classification accuracy of our

proposed RFNRC-TSK-MBE model is compared with ANFIS model in Table 8. The ANFIS system is trained

in 40 epochs, with two Gaussian membership functions for each of the five input variables. According to this

21

table, our proposed model outperforms ANFIS model in all the five investigated periods. Our proposed model

has reached to the average classification accuracy of 75.22% in five periods, whereas ANFIS could earn

classification accuracy of 65.06%, on average.

Please insert Table 8 about here.

Furthermore, the average appreciation in the stock price of the selected stocks (i.e., the stocks that are

classified as class 1 by the model) is compared with the average appreciation of all investigated stocks in the

subsequent year. Table 9 reports the experimental results in terms of average appreciation. Again, this table

confirms the ability of our proposed model for stock selection. The RFNRC-TSK-MBE could earn the excess

appreciation of 11.9%, 52.4%, 17.5%, 4.2%, 10.8% of the selected stocks over all 150 stocks in five consecutive

test periods and 19.36% on average.

Please insert Table 9 about here.

6. Conclusion

This paper proposes a new ensemble learning model to develop a modular TSK system for stock selection. The

proposed ensemble learning model includes four stages. The first stage involves data preprocessing and variable

selection. The second stage is about the data partitioning of the training data into several overlapping regions

using the proposed rough-fuzzy noise rejection clustering (RFNRC) algorithm. The proposed algorithm benefits

the strengths of rough, fuzzy and possibilistic clustering, while lacks their week points for developing a modular

system. The diversity of the individual learners is guaranteed by such a data partitioning algorithm. At the third

stage, an individual TSK system is generated for each region. The structure and parameter identification phases

of the TSK systems are done using Adeli-Hung algorithm and genetic algorithm. At the fourth stage, the outputs

of the individual TSK systems are aggregated using the proposed weighted ensemble strategy based on the

rough-fuzzy memberships.

Within this framework, while handling large data sets, each module may concentrate on knowledge

discovery within a different region of the problem. Subsequently, all of the modules contribute to problem-

solving with a degree based on the similarity of the instance with the prototypes of the modules. The similarity

is measured according to the rough partitions and the possibilistic-fuzzy memberships.

We implemented our proposed ensemble learning model on 150 Iranian companies with different activity

sectors listed on Tehran Stock Exchange (TSE) to develop a modular TSK system for stock selection. Based on

the experimental results, our developed modular system could appropriately select the stocks with information

22

coefficient of 18.15% on average, which is a good performance with respect to the previous researches [60, 88,

93]. Furthermore, our proposed system significantly outperformed the single TSK system (IC=8.5%) and also

other modular TSK systems, i.e., fuzzy noise rejection clustering based TSK system with averaging ensemble

(IC=12.74%) as well as rough-fuzzy noise rejection clustering based TSK system with averaging ensemble

(IC=15.24%). Investigating the comparison results, some conclusions can be drawn. First, modular systems

outperform a single system for problems with different regions of data characteristics, like stock selection

problem based on fundamental analysis. Second, the data partitioning using our proposed hybrid clustering

algorithm leads to the more capable modular systems. Additionally, considering the memberships in the

ensemble strategy improves the performance of the ensemble model, significantly.

Additionally, the performance of our proposed model is investigated for stock classification. According to

the results, our proposed model outperforms ANFIS regarding classification accuracy (75.22% Vs. 65.06%).

Based on this experiment, we can earn much more return on investment using the selected stocks by our

proposed model for portfolio diversification. The selected stocks reached to 19.36% excess appreciation on

average, where the average appreciation of all investigated stocks is 44.02%.

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Somayeh Mousavi received her B.Sc., M.Sc. and PhD degrees in Industrial Engineering from Amirkabir

University of Technology, Tehran, Iran in 2007, 2009 and 2015, respectively.

She is currently an Assistant Professor at Industrial Engineering Department, Meybod university, Meybod,

Yazd, Iran. Her main research interests include financial forecasting, portfolio selection, financial risk

management, fuzzy expert systems and applications of artificial intelligence and Meta-heuristics in financial

markets.

Dr. Mousavi has published her research articles on financial decision making using soft computing methods

in journals of Expert Systems with Applications, Knowledge based systems and Applied soft computing.

Akbar Esfahanipour received his B.Sc in Industrial Engineering from Amirkabir University of Technology,

Tehran, Iran in 1995. His M.Sc and PhD degrees are in industrial engineering from Tarbiat Modares University,

Tehran, Iran in 1998 and 2004, respectively. He is currently an Associate Professor at Industrial Engineering

Department, Amirkabir University of Technology. He has worked as a senior consultant for over 15 years in his

specialized field of expertise in various industries.

His research interests are in the areas of forecasting in financial markets, application of soft computing

methods in financial decision making, behavioral finance, financial resiliency, and analysis of financial risks.

Dr. Esfahanipour has published his research articles on financial decision making in prestigious journals such as

European Journal of Operational Research, Journal of Management Information Systems, Expert Systems with

Applications, Quantitative Finance, Knowledge-Based Systems, and Applied Soft Computing.

Mohammad Hossein Fazel Zarandi is Professor in Department of Industrial Engineering at Amirkabir

University of Technology, Tehran, Iran, and a member of the Knowledge-Information Systems Laboratory at

University of Toronto, Canada. His main research interests focus on Big Data Analytics, Artificial Intelligence,

29

Data Modeling, Soft Intelligent Computing, Deep Learning, Fuzzy Sets and Systems, Meta-heuristics, and

Optimization.

Professor Fazel Zarandi has published over 25 books and Book Chapters, more than 300 scientific

journal papers, more than 200 refereed conference papers and several technical reports in the above areas, most

of which are also accessible on the web. He has taught several courses in Big Data Analytics, Data Modeling,

Fuzzy Systems Engineering, Decision Support Systems, Information Systems, Artificial Intelligence and Expert

Systems, Systems Analysis and Design, Scheduling, Deep Learning, Simulations, and Multi-Agent Systems, at

several universities in Iran and North America.

30

Figure and table captions

Table 1. Position of this study among the related studies in the literature.

Table 2. Financial ratios as the possible input variables

Table 3. The training and testing periods of the five experiments

Table 4. Parameter settings of genetic algorithm

Table 5. Performance of the individual TSK systems on their corresponding training data subsets in terms of Information

Coefficient (IC)

Table 6. Performance of our proposed ensemble learning model and the other comparative models on testing periods in

terms of Information Coefficient (IC)

Table 7.Results of the student t-test for the pair wise comparison of the stock selection systems

Table 8. Classification performance of our proposed model versus ANFIS on testing periods in terms of classification*

accuracy

Table 9. Performance of our proposed RFNRC-TSK-MBE model on testing periods in terms of appreciation in selected

stocks price

Fig. 1. The overall framework of the proposed ensemble learning model for stock selection.

Fig. 2. Encoding the TSK consequent parameters as a GA chromosome.

Fig. 3. The module’s ensemble weights in a typical rough-fuzzy partitioning.

Fig. 4. Selection of the suitable weighting exponent.

Fig. 5. Identification of the optimum number of clusters.

Fig. 6. Application of the noise-rejection criterion.

31

Table 1. Position of this study among the related studies in the literature.

Reference Learning tool

No. of

fundamenta

l variables

Handle

nonlinearity

Fitness

measure

Type of

the

system

Consider different

fundamental

characteristics of

sectors

Becker et al. [60]

Genetic programming

65 Information coefficient

and spread

crisp

Quah [3] Artificial neural networks

11 Classification accuracy

Crisp and fuzzy

Huang et

al. [65]

Genetic

algorithm-

Fuzzy

12 Return of top

ranked

stocks

Fuzzy

Huang

[95]

support vector

regression-

genetic algorithms

14 Cumulative

return of

the selected stocks

Crisp

Vanstone

et al. [59]

Artificial neural

networks

4 Max.

percentage

change in price over

next 200

days

Crisp

Parque et

al. [61]

Genetic

network

programming

10 Risk-

adjusted

return of selected

stocks

Crisp

Ince [62] Genetic algorithm-Case-

based reasoning

7 Classification accuracy

Crisp

Yu et al.

[64]

Support Vector

Machines

20 Classificatio

n accuracy & the portfolio

accumulated

return

crisp

Silva et al.

[63]

Genetic

algorithm

10 Return and

risk of

proposed portfolio

Crisp

Shen &

Tzeng

[66]

VIKOR,

DANP,

DEMATEL (Decision-

making trial

and evaluation Laboratory)

17 Classificatio

n accuracy

fuzzy

This study Genetic

algorithm- Artificial neural

networks

36 Information

coefficient

TSK type

fuzzy rule-based

system

Table 2. Financial ratios as the possible input variables

Category Financial ratio

Profitability ratios Percentage of net profit to sale, percentage of operating profit to sale, percentage of gross profit to sale, percentage of gross margin to sale, percentage of net profit to gross margin, return on assets

(after tax), return on equity (after tax), return on working capital, working capital return percentage,

fixed assets return percentage

Liquidity ratios Current ratio, quick ratio, liquidity ratio, current assets ratio, networking capital

Activity ratios Inventory turnover, Average payment period, inventory to working capital, current assets turn over,

fixed asset turnover, total asset turnover

Leverage ratios Debt coverage ratio, debt to total assets ratio, debt to equity ratio, fixed assets to equity ratio, long-

term debt to equity ratio, current debt to equity ratio, equity ratio, interest coverage ratio

Valuation ratios Actual Earning per Share (EPS), net dividend per share, price to EPS ratio (P/E), book value per

share, dividend yield, price to book ratio(P/B), capitalization

32

Table 3. The training and testing periods of the five experiments

Experiment series Training period Testing period

Exp 1 Mar. 20, 1991- Mar. 19, 2009 Mar. 20, 2009- Mar. 19, 2010

Exp 2 Mar. 20, 1991- Mar. 19, 2010 Mar. 20, 2010- Mar. 19, 2011

Exp 3 Mar. 20, 1991- Mar. 19, 2011 Mar. 20, 2011- Mar. 19, 2012

Exp 4 Mar. 20, 1991- Mar. 19, 2012 Mar. 20, 2012- Mar. 19, 2013

Exp 5 Mar. 20, 1991- Mar. 19, 2013 Mar. 20, 2013- Mar. 19, 2014

Table 4. Parameter settings of genetic algorithm

Population size 100

Number of generations 200 Crossover rate 0.7

Mutation rate 0.3

Reproduction size 20

Table 5. Performance of the individual TSK systems on their corresponding training data subsets in terms of Information Coefficient (IC)

Training Subset Exp 1 Exp2 Exp3 Exp4 Exp5

U1 28.41% 28.11% 28.03% 27.67% 26.67%

U 2 40.70% 37.84% 30.64% 32.19% 30.45%

U 3 48.03% 45.45% 43.12% 41.60% 35.11%

U 4 33.01% 28.04% 31.00% 26.21% 28.35%

Notes: Ui is the ith training subset provided by the rough-fuzzy noise rejection clustering.

Table 6. Performance of our proposed ensemble learning model and the other comparative models on testing periods in terms of Information

Coefficient (IC)

Stock Selection system Exp 1 Exp 2 Exp 3 Exp 4 Exp 5 Average

Single TSK 6.51% 19.42% 1.81% 14.53% 0.26% 8.50%

FNRC-TSK-AveEa 7.51% 15.32% 15.54% 16.48% 8.86% 12.74%

RFNRC-TSK-AveEb 11.89% 18.06% 15.17% 17.85% 13.22% 15.24%

Our proposed system: RFNRC-

TSK-MBEc 12.43% 26.22% 19.86% 18.06% 14.16% 18.15%

aFNRC-TSK-AveE: fuzzy noise rejection clustering based TSK system with averaging ensemble bRFNRC-TSK-AveE: rough-fuzzy noise rejection clustering based TSK system with averaging ensemble cRFNRC-TSK-MBE: rough-fuzzy noise rejection clustering based modular TSK system with membership based ensemble

33

Table 7.Results of the student t-test for the pair wise comparison of the stock selection systems

Stock selection system RFNRC-TSK-AveE FNRC-TSK-AveE Single TSK

Our proposed system:

RFNRC-TSK-MBE

0.0012

[2.91%]

0.0000

[5.41%]

0.0000

[9.64%]

RFNRC-TSK-AveE

0.0027

[2.50%]

0.0000

[6.73%]

FNRC-TSK-AveE

0.0000

[4.24%]

Notes: The table reports the p-values of tests for the pair wise dominance of systems’ ICs. The difference between the IC averages of the respective systems are reported in

brackets

Table 8. Classification performance of our proposed model versus ANFIS on testing periods in terms of classification* accuracy

Model EXP1 EXP2 EXP3 EXP4 EXP5 Average

RFNRC-TSK-MBE 77.3% 80.9% 88.1% 79.5% 50.3% 75.22%

ANFIS 69.1% 72.6% 72% 61.9% 49.7% 65.06%

* In this experiment, all stocks have been classified in two classes. Class 1 represents stocks which appreciate in share price

equal or more than 80% within one year. Class 2 contains all other stocks.

Table 9. Performance of our proposed RFNRC-TSK-MBE model on testing periods in terms of appreciation in selected stocks price

EXP1 EXP2 EXP3 EXP4 EXP5 Average

Average appreciation of the selected stocks 67.2% 89.5% 19.7% 12.2% 128.3% 63.38%

Average appreciation of all 150 stocks 55.3% 37.1% 2.2% 8% 117.5% 44.02%

Excess appreciation of the selected stocks over all 150

stocks

11.9% 52.4% 17.5% 4.2% 10.8% 19.36%

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Fig. 1. The overall framework of the proposed ensemble learning model for stock selection.

a10 a11 …. a1n a20 a21 …. a2n …. ar0 ar1 … arn

Fig. 2. Encoding the TSK consequent parameters as a GA chromosome.

Fig. 3. The module’s ensemble weights in a typical rough-fuzzy partitioning.

TSK rule-based system1

TSK rule-based system2

TSK rule-based

systemC

.

.

.

Aggregating

the individual

systems based

on the

proposed

ensemble

strategy

Final

output

Fundamental

data

collection,

preparation

and

selection

Training data

partitioning

using the

proposed

rough-fuzzy

noise-

rejection

clustering

U2

.

.

.

UC

U1

35

Fig. 4. Selection of the suitable weighting exponent.

Fig. 5. Identification of the optimum number of clusters.

Fig. 6. Application of the noise-rejection criterion.